Boolean Algebra
Boolean algebra is like writing algebraic expressions acting on variables. Boolean notation is the set of symbols that define logical operators on variables.
$$ P = \text{NOT} (A \text{AND} B) P = \overline{A \cdot B} $$
$$ P = (A \text{AND} B) \text{OR} C P = (A \cdot B) + C $$
NOT
$$ P = \text{NOT} A P = \overline{A} $$
What does the notation $\overline{A}$ in boolean algebra??
NOT.
AND
$$ P = A \text{AND} B P = A \cdot B $$
What does the notation $A \cdot B$ mean in boolean algebrea??
AND.
OR
$$ P = A \text{OR} B P = A + B $$
What does the notation $A + B$ mean in boolean algebra??
OR.
XOR
$$ P = A \text{XOR} B P = A \oplus B $$
What does the notation $A \oplus B$ mean in boolean algebra??
XOR.
NOR and NAND
Instead of having a special notation, you write these as boolean expressions themselves.
$$ P = \text{NOT} (A \text{OR} B) P = \overline{(A + B)} $$
What is NOR in boolean notation??
$$ \overline{(A+B)} $$
What is NAND in boolean notation??
$$ \overline{(A \cdot B)} $$
What is the order of operations for boolean algebra??
- Highest: NOT
- Middle: AND
- Lowest: OR
De Morgan’s Laws
Who was Augustus De Morgan??
August De Morgan was a mathematician who invented laws to simplify boolean expressions.
What is De Morgan’s first law??
$$ \overline{A} \cdot \overline{B} = \overline{A+B} $$
What is $\overline{A} \cdot \overline{B}$ equivalent to??
$$ \overline{A + B} $$
What is De Morgan’s second law??
$$ \overline{A \cdot B} = \overline{A} + \overline{B} $$
What is $\overline{A \cdot B}$ equivalent to??
$$ \overline{A} + \overline{B} $$
In boolean algebra, simplify $X \cdot 0$$??
$0$
In boolean algebra, simplify $X \cdot 1$??
$X$
In boolean algebra, simplify $X \cdot X$??
$X$
In boolean algebra, simplify $X \cdot \overline{X}$??
$0$
In boolean algebra, simplify $X + 1$??
$X$
In boolean algebra, simplify $X + 1$??
$1$
In boolean algebra, simplify $X + X$??
$X$
In boolean algebra, simplify $X + \overline{X}$??
$1$
In boolean algebra, simplify $\overline{\overline{X}}$??
$X$
What is the commutative rule??
The order of operations does not matter.
Because of the commutative rule, what is $X \cdot Y$ equivalent to??
$$ Y \cdot X $$
What is the associative rule??
Doing A then B is the same as doing B then A.
Because of the associative rule, what is $X \cdot (Y \cdot Z)$ equivalent to??
$$ (X \cdot Y) \cdot Z $$
What is the distributive rule??
Applying an operand to a bracket is the same as applying the operand to each term of the bracket.
Because of the distributive rule, what is $X \cdot (Y + Z)$ equivalent to??
$$ X \cdot Y + X \cdot Z $$
In boolean algebra, simplify $(A \cdot \overline{A}) + B$??
$$ B $$
In boolean algebra, simplify $(A \cdot B) + (\overline{A} \cdot B)$??
$$ B $$
In boolean algebra, simplify $A \cdot B + A \cdot (B + C)$??
$$ A \cdot (B + C) $$
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date: 2020-09-08 10:08
tags:
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- '@?school'
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title: Computing - Boolean Algebra